3.25 \(\int \frac{A+B x}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b}}+\frac{B \sqrt{a+b x^2}}{b} \]

[Out]

(B*Sqrt[a + b*x^2])/b + (A*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]

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Rubi [A]  time = 0.0149365, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {641, 217, 206} \[ \frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b}}+\frac{B \sqrt{a+b x^2}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/Sqrt[a + b*x^2],x]

[Out]

(B*Sqrt[a + b*x^2])/b + (A*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{A+B x}{\sqrt{a+b x^2}} \, dx &=\frac{B \sqrt{a+b x^2}}{b}+A \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{B \sqrt{a+b x^2}}{b}+A \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{B \sqrt{a+b x^2}}{b}+\frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0294159, size = 46, normalized size = 1.07 \[ \frac{A \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{\sqrt{b}}+\frac{B \sqrt{a+b x^2}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/Sqrt[a + b*x^2],x]

[Out]

(B*Sqrt[a + b*x^2])/b + (A*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/Sqrt[b]

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Maple [A]  time = 0.004, size = 37, normalized size = 0.9 \begin{align*}{\frac{B}{b}\sqrt{b{x}^{2}+a}}+{A\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x^2+a)^(1/2),x)

[Out]

B*(b*x^2+a)^(1/2)/b+A*ln(x*b^(1/2)+(b*x^2+a)^(1/2))/b^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77601, size = 223, normalized size = 5.19 \begin{align*} \left [\frac{A \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \, \sqrt{b x^{2} + a} B}{2 \, b}, -\frac{A \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) - \sqrt{b x^{2} + a} B}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[1/2*(A*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*sqrt(b*x^2 + a)*B)/b, -(A*sqrt(-b)*arctan(
sqrt(-b)*x/sqrt(b*x^2 + a)) - sqrt(b*x^2 + a)*B)/b]

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Sympy [B]  time = 1.01588, size = 102, normalized size = 2.37 \begin{align*} A \left (\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (x \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}\right ) + B \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: b = 0 \\\frac{\sqrt{a + b x^{2}}}{b} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x**2+a)**(1/2),x)

[Out]

A*Piecewise((sqrt(-a/b)*asin(x*sqrt(-b/a))/sqrt(a), (a > 0) & (b < 0)), (sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a),
(a > 0) & (b > 0)), (sqrt(-a/b)*acosh(x*sqrt(-b/a))/sqrt(-a), (b > 0) & (a < 0))) + B*Piecewise((x**2/(2*sqrt
(a)), Eq(b, 0)), (sqrt(a + b*x**2)/b, True))

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Giac [A]  time = 1.18092, size = 53, normalized size = 1.23 \begin{align*} -\frac{A \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{\sqrt{b}} + \frac{\sqrt{b x^{2} + a} B}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

-A*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + sqrt(b*x^2 + a)*B/b